Blast Into Math!

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About the book

Reviews

lokender
★★★★★

Jemil Alvarez
★★★★★

I think the book is very clear and that it strengthens those who don't understand math.

Sorin Neagu-Ventzel
★★★★★

20 August 2014

Excellent one! I will use it for sure with my students who are preparing for math contests. Thanks!

Chanaka Sudheera
★★★★★

4 June 2014

A very useful textbook!

Zohaib Nasir
★★★★★

28 March 2014

I think by reading this anyone can increase their ability to solve math problems. :-)

Description

Blast into Math! A fun rigorous introduction to pure mathematics which is suitable for both students and a general audience interested in learning what pure mathematics is all about. Pure mathematics is presented in a friendly, accessible, and nonetheless rigorous style. Definitions, theorems, and proofs are accompanied by creative analogies and illustrations to convey the meaning and intuition behind the abstract math. The key to reading and understanding this book is doing the exercises. You don't need much background for the first few chapters, but the material builds upon itself, and if you don't do the exercises, eventually you'll have trouble understanding. The book begins by introducing fundamental concepts in logic and continues on to set theory and basic topics in number theory. The sixth chapter shows how we can change our mathematical perspective by writing numbers in bases other than the usual base 10. The last chapter introduces analysis. Readers will be both challenged and encouraged. A parallel is drawn between the process of working through the book and the process of mathematics research. If you read this book and do all the exercises, you will not only learn how to prove theorems, you'll also experience what mathematics research is like: exciting, challenging, and fun!

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Content

Preface

To the reader

Pure mathematics: the proof of the pudding is in the eating

A universal language

Theorems, propositions, and lemmas

Logic

Ready? Set? Prove!

Exercises

Examples and hints

Sets of numbers: mathematical playgrounds

Set theory

Numbers

The least upper bound property

Proof by induction

Exercises

Examples and hints

The Euclidean algorithm: a computational recipe

Division

Greatest common divisors

Proof of the Euclidean Algorithm

Greatest common divisors in disguise

Exercises

Examples and hints

Prime numbers: indestructible building blocks

Ingredients in the proof of the Fundamental Theorem of Arithmetic

Unique prime factorization: the Fundamental Theorem of Arithmetic

How many primes are there?

Counting infinity

Exercises

Examples and hints

Mathematical perspectives: all your base are belong to us

Number bases: infinitely many mathematical perspectives

Fractions in bases

Exercises

Examples and hints

Analytic number theory: ants, ghosts and giants

Sequences: mathematical ants

Real numbers and friendly rational numbers

Series: a tower of mathematical ants

Decimal expansions

The Prime Number Theorem

Exercises

Examples and hints

Afterword

Bibliography

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